Question

Let Tetra act on the set S of pairs of opposite edges of the tetra- hedron. What is the stabilizer of one such pair? Use Corollary 1.4 to recompute |Tetral. 1. Corollary 1.4. If a finite group G acts on a set S, then for any SES, IG| = #(Os)|G5|. In particular, # (0s) divides |G|.

Answer 1

For the symmetry group of a tetrahedron, we have;

action	#orbit	#stab	|G|
on edges	6	2	12

It is to be noted that here its a bit tricky to find the stabilizer of an edge but since we know there are 2 elements in the stabilizer from the orbit-stabilizer theorem.